Question

four students wrote sequences during math class.
andre: $-\frac{3}{4}, \frac{3}{8}, -\frac{3}{16}, -\frac{3}{32}, \dots$
brenda: $\frac{3}{4}, -\frac{3}{8}, \frac{3}{16}, \frac{3}{32}, \dots$
camille: $\frac{3}{4}, \frac{3}{8}, -\frac{3}{16}, -\frac{3}{32}, \dots$
doug: $\frac{3}{4}, -\frac{3}{8}, \frac{3}{16}, -\frac{3}{32}, \dots$
which student wrote a geometric sequence?
options:
andre
brenda
camille
doug

Explanation:

Step1: Recall Geometric Sequence Definition

A geometric sequence has a common ratio \( r \), where \( r=\frac{a_{n + 1}}{a_n} \) for all \( n \).

Step2: Analyze Andre's Sequence

Andre's sequence: \( -\frac{3}{4},\frac{3}{8},-\frac{3}{16},-\frac{3}{32},\dots \)
Calculate \( r_1=\frac{\frac{3}{8}}{-\frac{3}{4}}=\frac{3}{8}\times(-\frac{4}{3})=-\frac{1}{2} \)
\( r_2=\frac{-\frac{3}{16}}{\frac{3}{8}}=-\frac{3}{16}\times\frac{8}{3}=-\frac{1}{2} \)? Wait, no: \( \frac{-\frac{3}{16}}{\frac{3}{8}}=-\frac{3}{16}\times\frac{8}{3}=-\frac{1}{2} \), but next \( r_3=\frac{-\frac{3}{32}}{-\frac{3}{16}}=\frac{-\frac{3}{32}}{-\frac{3}{16}}=\frac{1}{2} \). Not constant.

Step3: Analyze Brenda's Sequence

Brenda's sequence: \( \frac{3}{4},-\frac{3}{8},\frac{3}{16},\frac{3}{32},\dots \)
\( r_1=\frac{-\frac{3}{8}}{\frac{3}{4}}=-\frac{1}{2} \), \( r_2=\frac{\frac{3}{16}}{-\frac{3}{8}}=-\frac{1}{2} \), \( r_3=\frac{\frac{3}{32}}{\frac{3}{16}}=\frac{1}{2} \). Not constant.

Step4: Analyze Camille's Sequence

Camille's sequence: \( \frac{3}{4},\frac{3}{8},-\frac{3}{16},-\frac{3}{32},\dots \)
\( r_1=\frac{\frac{3}{8}}{\frac{3}{4}}=\frac{1}{2} \), \( r_2=\frac{-\frac{3}{16}}{\frac{3}{8}}=-\frac{1}{2} \). Not constant.

Step5: Analyze Doug's Sequence

Doug's sequence: \( \frac{3}{4},-\frac{3}{8},\frac{3}{16},-\frac{3}{32},\dots \)
Calculate \( r_1=\frac{-\frac{3}{8}}{\frac{3}{4}}=-\frac{3}{8}\times\frac{4}{3}=-\frac{1}{2} \)
\( r_2=\frac{\frac{3}{16}}{-\frac{3}{8}}=\frac{3}{16}\times(-\frac{8}{3})=-\frac{1}{2} \)
\( r_3=\frac{-\frac{3}{32}}{\frac{3}{16}}=-\frac{3}{32}\times\frac{16}{3}=-\frac{1}{2} \)
Common ratio \( r = -\frac{1}{2} \) (constant).

Answer:

Doug